Consider the valve opening data displayed in Table 11.1. Suppose the data represent valves produced on four different machines on three different shifts and that the quality controllers want to know whether there is any difference in the mean measurements of valve openings by shift or by machine. The data are given here, organized by machine and shift. In addition, Excel has been used to analyze the data with a two-way ANOVA. What are the hypotheses for this problem? Study the output in terms of significant differences. Discuss the results obtained. What conclusions might the quality controllers reach from thisanalysis?

Valve Openings (cm) Shift 1 Shift 2 Shift 3 6.56 6.38 6.40 6.19 6.54 6.26 6.34 6.23 6.58 6.22 6.44 6.27 6.36 6.29 6.50 6.19 6.29 6.23 6.19 6.33 6.26 6.31 6.21 6.58 Machine 4 ANOVA: Two-Factor with Replication ANOVA Source of Variation df MS F P-value F crit Sample Columns Interaction Within 0.00538 3 0.00179 0.14 0.9368 3.49 0.19731 20.09865 7.47 0.0078 3.89 0.03036 6 0.00506 0.38 0.8760 3.00 0.15845 12 0.01320 0.39150 23